In mathematics, the Christoffel–Darboux formula or Christoffel–Darboux theorem is an identity for a sequence of orthogonal polynomials, introduced by Elwin Bruno Christoffel (1858) and Jean Gaston Darboux (1878).
Christoffel–Darboux formula — if a sequence of polynomials
, and orthogonal with respect to a probability measure
are the squared norms, and
are the leading coefficients.
There is also a "confluent form" of this identity by taking
limit: Christoffel–Darboux formula, confluent form —
be a sequence of polynomials orthonormal with respect to a probability measure
(they are called the "Jacobi parameters"), then we have the three-term recurrence[1]
is a linear combination of
, it suffices to perform Gram-Schmidt process on
, which yields the desired recurrence.
For any sequence of nonzero constants
, and both sides of the equation would remain unchanged.
Thus WLOG, scale each
polynomial, it is perpendicular to
Base case:
By the three-term recurrence,
The Hermite polynomials are orthogonal with respect to the gaussian distribution.
polynomials are orthogonal with respect to
polynomials are orthogonal with respect to
are orthonormal with respect to the exponential distribution
Associated Legendre polynomials: The summation involved in the Christoffel–Darboux formula is invariant by scaling the polynomials with nonzero constants.
Thus, each probability distribution
defines a series of functions
which are called the Christoffel–Darboux kernels.
By the orthogonality, the kernel satisfies
In other words, the kernel is an integral operator that orthogonally projects each polynomial to the space of polynomials of degree up to
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